So, what we’ll do is restrict ourselves to the
region 1 ≤ x < 4. The idea behind this remains the same – the equation will
‘retain’ its form in this region. That is, |x – 4| will always equal 4 – x and
|x – 1| will always equal x – 1.

Let’s solve the equation then.

|x – 1| + |x – 4| = 7 implies x – 1 + 4 – x =
7, which gives 3 = 7.

Well, that’s strange. Can you make sense of
it?

This means that when x lies between 1 and 4,
the LHS will always be 3.

To solve an equation of the form |x – 1| + |x
– 4| = 7, we divided the number line into some ‘regions’, such that in a
particular region, the expressions inside the modulus retain their signs. Or,
|x – 4| remained x – 4 (or 4 – x) throughout that region.